Sanskrit commentators have a technical term for it: kalpa-bheda, difference of kalpa. Where two Purāṇic accounts of an episode conflict, the accounts are taken to belong to different world-cycles. They are not rival reports competing for one slot; they are the same story told of different turns of the wheel. The device is not applied ad hoc by later apologists alone — the texts are built for it. The Viṣṇu Purāṇa (III.3) enumerates twenty-eight Vyāsas, one for each dvāpara-yuga. Vyāsa is an office, not a man, so the dictation of the Mahābhārata is itself a recurrent event, and asking which telling is the real one is a category error inside that frame.
Formally the manoeuvre is the same in every case where it appears. A set of sentences that is jointly unsatisfiable is replaced by a family of sentences indexed by a parameter, and each member of the family is asserted only at its own index. What looked like S ∧ ¬S becomes S(k₁) ∧ ¬S(k₂), which is consistent whenever k₁ ≠ k₂. Contradiction is a property of statements, not of the world; restoring a suppressed index can remove it.
A film strip is the everyday form of the same object: the sprocket axis is the base, and a frame hangs at each position along it. Nothing travels along the strip, and the whole strip exists at once — motion is what appears when it is read with an index. Choosing one account per index is, in the same vocabulary, choosing a section.
An index is source-supplied when the source names it before the discrepancy is noticed — the text says which kalpa, the theory says which vacuum. It is reader-supplied when the interpreter introduces it afterwards, to absorb a conflict already in view. The two look identical on the page and behave completely differently.
Block [1] of the companion script enumerates every proposition over two variables that is neither unsatisfiable nor tautologous — fourteen of them — and examines all 455 sets of size two or three. Of those, 255 are jointly unsatisfiable. A reader free to assign an index after seeing the sentences rescues all 255. Model
Making the index continuous rather than enumerable sends K → ∞. At N = 20 the chance baseline is 0.9094 at K = 103, 0.99991 at 106 and 0.9999999 at 109. Two points drawn from a continuum never coincide, so no collision is ever forced. An index set that is a space rather than a list forbids nothing at all: enriching the index is the one move that guarantees it cannot bite.
Common descent behaves identically. Any two things share an ancestor if the search is allowed to go deep enough, so an unqualified claim of shared origin holds universally and separates nothing — the K → ∞ case again, with depth standing in for index size. The informative version always names the depth: philology names the ancestor copy, biology names and dates the last common ancestor, cosmology names nucleosynthesis or recombination. A shared origin with no depth attached forbids nothing.
The result is a triviality, and the triviality is the finding. Give each sentence its own index and the only sets that resist are the ones containing a sentence that is unsatisfiable on its own — which is to say, the ones where nothing was ever being described. The construction takes one line and works universally.
A test that cannot fail returns no information when it passes. This is the same objection the defect ledger makes to a proof whose hypotheses are never checked, and the same one the null-result papers in this volume make to a fit that would have come out the same on noise. Observing that an indexed reading is available tells a reader nothing, because it is always available.
A source-supplied index behaves differently because it is committed in advance and can collide with itself. Block [2] states the chance baseline. Let N sentences bear on one question; each asserts it or denies it, and each carries a label drawn from the K the source provides. The indexed reading survives only if no single label carries both an assertion and a denial. Conditioning on the number of labels actually used gives a closed form, evaluated exactly:
with S(N, m) the Stirling numbers of the second kind. Every term is positive, so the evaluation is stable as well as exact; it is checked against hand calculation at three points. For twenty sentences bearing on one question:
| K — labels the source provides | P(survives by chance) | |
|---|---|---|
| 8 | 0.0002 | a committed index; collisions are forced |
| 20 | 0.0111 | one label per statement — still bites |
| 50 | 0.1508 | |
| 138 | 0.5000 | half the discriminating power gone |
| 400 | 0.7885 | |
| 1853 | 0.9500 | indistinguishable from no index at all |
Block [3] repeats the calculation at N = 10, 20, 30 and 40 and finds the half-power point scaling as the square of the number of statements, with a flat constant:
The mechanism is the birthday collision: there are about N²/2 pairs, each shares a label with probability 1/K, and half of those disagree in sign. So the expected number of forced collisions is N²/4K, and the index stops discriminating once K reaches order N². Model
Legitimacy of provenance is necessary and not sufficient. An index the source genuinely supplies still explains nothing once the source is willing to supply enough of them. The quantity that matters is not who named the index but how large the index set is relative to the number of statements it is being asked to reconcile.
String compactification is the case where the two senses of the word coincide: the extra dimensions really are a fibre, a Calabi–Yau sitting over each point of spacetime. But the plurality of worlds does not live in those dimensions. It lives in the choice — which manifold, which flux — and that choice is an index. The dimensions are the same six everywhere. Literature
The quoted sizes of that index set are 10500 for the flux-vacua estimate and far larger, around 10272000, in F-theory flux counting. Block [4] puts them against the blunting scale. The Standard Model has nineteen free parameters, twenty-six once neutrino masses and mixing are included, so the scale at which an index stops discriminating is of order 102.
Hundreds of orders of magnitude past the point where the chance baseline reaches one. This is not a new objection — it is the standard motivation for the swampland programme, restated in the terms of block [3]. That programme is the attempt to show the assignment is not by chance: to find effective theories that get no label, so that the index excludes something after all. Its existence is the concession that size alone had blunted it. Open
The parallel is offered as a parallel and not as support. String theory is without experimental confirmation, and a humanistic reading claim that leaned on it for authority would be borrowing from an account that is itself unsettled. What the two cases share is the demand, arrived at independently and a thousand years apart: name the index, and show that it forbids something.
General relativity supplies a worked instance in which did the source supply the index? has a computed answer rather than an argued one. Cutting a spacetime into frames is choosing a foliation. A generic spacetime supplies none: by the relativity of simultaneity, different observers slice the same block differently and no slicing is the correct one. There the index is reader-supplied, and the question of which slice is now has no answer to give.
A spacetime is static exactly when it admits a hypersurface-orthogonal timelike Killing vector. Where one exists, the geometry hands over the slicing itself and the index is source-supplied. So in this case the criterion of section 5 reduces to the existence of a Killing field — a condition one computes. Literature
The distinction that makes this work is worth keeping separate from the picture that motivates it. A slice is frozen because it contains no time. The block is not frozen at all: it contains the whole of time, which is why it needs no second time to change in. A layered account of the world does not abolish the index — it hands the index the entire job, because layers with no ordering on them are not a history of anything.
The index has to come from the source, not the reader. Where the source names it before the discrepancy is in view, read variants as indexed. Where it does not, a discrepancy stays a discrepancy until evidence supplies one.
And the index set has to be small. Against N statements, an index set of order N² or larger has no discriminating power left, whoever supplied it. An indexed reading earns its keep only by forbidding combinations that could have occurred.
And the index has to be attested before the discrepancy it settles. Presence in the source is not sufficient. A pedigree attached once the conflict is already in view answers the source supplied it while doing the work of a reader. The standing case is Ayurveda's own descent from Brahmā through the Aśvins to Indra, recited at the head of both saṁhitās: Zysk's reading places the empirical medicine in the śramaṇa and Buddhist ascetic milieu with the Vedic lineage fitted afterwards. Whether that reading is correct is a question for textual criticism — which is exactly the point. Attestation order is datable; provenance on its own is not.
Applied to the case that prompted the paper: kalpa-bheda is source-supplied, and correctly so. It is the tradition's own device, named in its own vocabulary, and the enumerated Vyāsas show the index was in the architecture before any commentator needed it. Whether it also passes the second test is an empirical question about a specific corpus — how many cycles are actually named against how many variants are actually reconciled — and this paper does not answer it. It states what answering it would require.
The rule fails if a corpus can be exhibited whose index is supplied only by its readers and which nonetheless makes checkable predictions that could have come out otherwise. None is offered here. The rule is stated so that such a corpus would count against it.